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数理解析研究所講究録

Correlations of measurement information and noise in quantum measurements with finite resolution (Analytical Study of Quantum Information and Related Fields)

Holger Friedrich Hofmann, Takayoshi Kobayashi, Akira Furusawa

Published Apr 1, 2001• 0 citations
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Abstract

The original purpose of measurements is to provide us with information about a previously unknown physical property of the systeIn observed.In the Hilbert space formalism of quantum mechanics, this physical meaning of measurement information is not immediately apparent.In order to study the relationship between the Hilbert space coherence of the quantum state and the measurement information obtained in the laboratory, we introduce a generalized measurement postulate for finite resolution measurements.With this measurement model, correlations between non-commuting observables can be investigated.These experimentally accessible correlations reveal nonclassical features in their dependence on the operator ordering, reflecting the particular measurement context by which they are determined.

Related papers

Correlations of measurement information and noise in quantum measurements with finite resolution

Holger Friedrich Hofmann, Takayoshi Kobayashi, Akira Furusawa

The original purpose of measurements is to provide us with information about a previously unknown physical property of the system observed. In the Hilbert space formalism of quantum mechanics, this physical meaning of measurement information is not immediately apparent. In order to study the relationship between the Hilbert space coherence of the quantum state and the measurement information obtained in the laboratory, we introduce a generalized measurement postulate for finite resolution measurements. With this measurement model, correlations between non-commuting observables can be investigated. These experimentally accessible correlations reveal nonclassical features in their dependence on the operator ordering, reflecting the particular measurement context by which they are determined.

arXiv:quant-ph/0003067•Mar 17, 2000•0 citations

Uncertainty characteristics of generalized quantum measurements

Holger Friedrich Hofmann

The effects of any quantum measurement can be described by a collection of measurement operators ${{M}_{m}}$ acting on the quantum state of the measured system. However, the Hilbert space formalism tends to obscure the relationship between the measurement results and the physical properties of the measured system. In this paper, a characterization of measurement operators in terms of measurement resolution and disturbance is developed. It is then possible to formulate uncertainty relations for the measurement process that are valid for arbitrary input states. The motivation of these concepts is explained from a quantum communication viewpoint. It is shown that the intuitive interpretation of uncertainty as a relation between measurement resolution and disturbance provides a valid description of measurement back action. Possible applications to quantum cryptography, quantum cloning, and teleportation are discussed.

arXiv:quant-ph/0209162•Physical Review A•Feb 13, 2003•18 citations

Correlation properties of quantum measurements

Paul Busch, Pekka J. Lahti

The kind of information provided by a measurement is determined in terms of the correlation established between observables of the apparatus and the measured system. Using the framework of quantum measurement theory, necessary and sufficient conditions for a measurement interaction to produce strong correlations are given and are found to be related to properties of the final object and apparatus states. These general results are illustrated with reference to the standard model of the quantum theory measurement.

arXiv:quant-ph/9603016•Journal of Mathematical Physics•Jun 1, 1996•13 citations

Information as Structural Tension: The Φ-Conjecture

Claudio Menéndez

Title:Information as Structural Tension: The Φ-Conjecture Author:Claudio A. Menéndez (ORCID: 0009-0001-8869-614X) Description:This paper formulates and develops the Φ–Information Conjecture — a structural and mathematically precise proposal that reinterprets information not as a fundamental physical entity, but as an emergent residue arising from the incompatibility between two rational orders: a spectral rational order (selection of coherent modes) and a geometric resolution order (coarse–graining at finite scale). Formally, for a Hilbert space HH and two operators PNPN (spectral projection) and GℓGℓ (resolution operator), the Φ–residue is defined as ΔΦ(ℓ)=PNGℓ−GℓPN,ΔΦ(ℓ)=PNGℓ−GℓPN, and the Φ–information at scale ℓℓ is given by its operator norm: IΦ(ℓ)=∥ΔΦ(ℓ)∥.IΦ(ℓ)=∥ΔΦ(ℓ)∥. The conjecture is developed within a minimal operator–algebraic framework, free of ontological commitments, and is illustrated through explicit finite–dimensional examples (including two– and three–dimensional models) that exhibit nontrivial scaling behavior. Basic structural properties are proven, a minimal Heisenberg–type dynamics for the residue is introduced, and conceptual bridges to Shannon entropy, von Neumann entropy, and quantum measurement incompatibility are outlined. A dedicated discussion section addresses potential objections regarding tautology, parameter choices, and the physical meaning of scale dependence, clarifying the conjecture’s role as a structural — rather than ontological — proposal about the nature of information. Keywords:Information Theory; Non–Commutative Residue; Rational Orders; Spectral Projection; Coarse–Graining; Operator Algebras; Epistemic Structuralism; Shannon Entropy; Von Neumann Entropy; Quantum Measurement; Finite–Resolution Physics. Related Identifiers: Continues: The Φ-Model: Epistemic Noncommutativity and Finite-Resolution Physics Integrates: The Rational Circle: Finite-Resolution Measurement and the Emergence of Mathematical Constants Integrates: Time as a Projection: Observational Consequences and Empirical Tests of the Φ-Model Extends: Development of Computational Tools for Numerical Predictions of the Φ-Model Related: Finite-Resolution Quantum Geometry — Linking Non-Commutative Projection with Loop Quantum Gravity Related: Numerical Implementation and Extended Analysis of Finite-Resolution Quantum Geometry — Computational Verification of the Δℓ Framework Related: The Emergent Geometry Conjecture — A Research Program for Classical Spacetime from Quantum Algebras Related: The Φ-Model Companion: A Unified Framework of Finite Coherence Scientific Context:This work contributes to the foundations of information theory by proposing a scale–dependent, operator–theoretic definition of information grounded in the non–commutativity of rational filters. It aligns with the broader Φ–Model research program — which seeks to describe physical reality through finite–resolution projections of underlying rational structures — while remaining self–contained and accessible to mathematicians, theoretical physicists, and philosophers of science.

•Nov 13, 2025•0 citations

Information and noise in quantum measurement

Holger Friedrich Hofmann

Even though measurement results obtained in the real world are generally both noisy and continuous, quantum measurement theory tends to emphasize the ideal limit of perfect precision and quantized measurement results. In this article, a more general concept of noisy measurements is applied to investigate the role of quantum noise in the measurement process. In particular, it is shown that the effects of quantum noise can be separated from the effects of information obtained in the measurement. However, quantum noise is required to ``cover up'' negative probabilities arising as the quantum limit is approached. These negative probabilities represent fundamental quantum-mechanical correlations between the measured variable and the variables affected by quantum noise.

arXiv:quant-ph/9907018•Physical Review A•Jul 6, 2000•12 citations

Weak valued statistics as fundamental explanation of quantum physics

Holger Friedrich Hofmann

Recently, weak measurements have attracted a lot of interest as an experimental method for the investigation of non-classical correlations between observables that cannot be measured jointly. Here, I explain how the complex valued statistics observed in weak measurements relate to the operator algebra of the conventional Hilbert space formalism and show that the algebra of operators originates from more fundamental relations between the physical properties of a quantum system that can be expressed in terms of complex conditional probabilities. In particular, commutation relations can be identified with fundamental imaginary correlations that characterize the relations between physical properties in terms of their transformation dynamics. Non-commutativity thus originates from a definition of relations between physical properties that replaces the assumption of joint reality with a complex-valued probability reflecting the dynamical response of the system to external forces, e.g. in measurement interactions.

arXiv:1305.0053•Apr 30, 2013•1 citation
The original purpose of measurements is to provide us with information about a previously unknown physical property of the systeIn observed.In the Hilbert space formalism of quantum mechanics, this physical meaning of measurement information is not immediately apparent.In order to study the relationship between the Hilbert space coherence of the quantum state and the measurement information obtained in the laboratory, we introduce a generalized measurement postulate for finite resolution measurements.With this measurement model, correlations between non-commuting observables can be investigated.These experimentally accessible correlations reveal nonclassical features in their dependence on the operator ordering, reflecting the particular measurement context by which they are determined.
The original purpose of measurements is to provide us with information about a previously unknown physical property of the system observed. In the Hilbert space formalism of quantum mechanics, this physical meaning of measurement information is not immediately apparent. In order to study the relationship between the Hilbert space coherence of the quantum state and the measurement information obtained in the laboratory, we introduce a generalized measurement postulate for finite resolution measurements. With this measurement model, correlations between non-commuting observables can be investigated. These experimentally accessible correlations reveal nonclassical features in their dependence on the operator ordering, reflecting the particular measurement context by which they are determined.
The effects of any quantum measurement can be described by a collection of measurement operators Mm{{M}_{m}}Mm​ acting on the quantum state of the measured system. However, the Hilbert space formalism tends to obscure the relationship between the measurement results and the physical properties of the measured system. In this paper, a characterization of measurement operators in terms of measurement resolution and disturbance is developed. It is then possible to formulate uncertainty relations for the measurement process that are valid for arbitrary input states. The motivation of these concepts is explained from a quantum communication viewpoint. It is shown that the intuitive interpretation of uncertainty as a relation between measurement resolution and disturbance provides a valid description of measurement back action. Possible applications to quantum cryptography, quantum cloning, and teleportation are discussed.
The kind of information provided by a measurement is determined in terms of the correlation established between observables of the apparatus and the measured system. Using the framework of quantum measurement theory, necessary and sufficient conditions for a measurement interaction to produce strong correlations are given and are found to be related to properties of the final object and apparatus states. These general results are illustrated with reference to the standard model of the quantum theory measurement.
Title:Information as Structural Tension: The Φ-Conjecture Author:Claudio A. Menéndez (ORCID: 0009-0001-8869-614X) Description:This paper formulates and develops the Φ–Information Conjecture — a structural and mathematically precise proposal that reinterprets information not as a fundamental physical entity, but as an emergent residue arising from the incompatibility between two rational orders: a spectral rational order (selection of coherent modes) and a geometric resolution order (coarse–graining at finite scale). Formally, for a Hilbert space HH and two operators PNPN (spectral projection) and GℓGℓ (resolution operator), the Φ–residue is defined as ΔΦ(ℓ)=PNGℓ−GℓPN,ΔΦ(ℓ)=PNGℓ−GℓPN, and the Φ–information at scale ℓℓ is given by its operator norm: IΦ(ℓ)=∥ΔΦ(ℓ)∥.IΦ(ℓ)=∥ΔΦ(ℓ)∥. The conjecture is developed within a minimal operator–algebraic framework, free of ontological commitments, and is illustrated through explicit finite–dimensional examples (including two– and three–dimensional models) that exhibit nontrivial scaling behavior. Basic structural properties are proven, a minimal Heisenberg–type dynamics for the residue is introduced, and conceptual bridges to Shannon entropy, von Neumann entropy, and quantum measurement incompatibility are outlined. A dedicated discussion section addresses potential objections regarding tautology, parameter choices, and the physical meaning of scale dependence, clarifying the conjecture’s role as a structural — rather than ontological — proposal about the nature of information. Keywords:Information Theory; Non–Commutative Residue; Rational Orders; Spectral Projection; Coarse–Graining; Operator Algebras; Epistemic Structuralism; Shannon Entropy; Von Neumann Entropy; Quantum Measurement; Finite–Resolution Physics. Related Identifiers: Continues: The Φ-Model: Epistemic Noncommutativity and Finite-Resolution Physics Integrates: The Rational Circle: Finite-Resolution Measurement and the Emergence of Mathematical Constants Integrates: Time as a Projection: Observational Consequences and Empirical Tests of the Φ-Model Extends: Development of Computational Tools for Numerical Predictions of the Φ-Model Related: Finite-Resolution Quantum Geometry — Linking Non-Commutative Projection with Loop Quantum Gravity Related: Numerical Implementation and Extended Analysis of Finite-Resolution Quantum Geometry — Computational Verification of the Δℓ Framework Related: The Emergent Geometry Conjecture — A Research Program for Classical Spacetime from Quantum Algebras Related: The Φ-Model Companion: A Unified Framework of Finite Coherence Scientific Context:This work contributes to the foundations of information theory by proposing a scale–dependent, operator–theoretic definition of information grounded in the non–commutativity of rational filters. It aligns with the broader Φ–Model research program — which seeks to describe physical reality through finite–resolution projections of underlying rational structures — while remaining self–contained and accessible to mathematicians, theoretical physicists, and philosophers of science.
Even though measurement results obtained in the real world are generally both noisy and continuous, quantum measurement theory tends to emphasize the ideal limit of perfect precision and quantized measurement results. In this article, a more general concept of noisy measurements is applied to investigate the role of quantum noise in the measurement process. In particular, it is shown that the effects of quantum noise can be separated from the effects of information obtained in the measurement. However, quantum noise is required to ``cover up'' negative probabilities arising as the quantum limit is approached. These negative probabilities represent fundamental quantum-mechanical correlations between the measured variable and the variables affected by quantum noise.
Recently, weak measurements have attracted a lot of interest as an experimental method for the investigation of non-classical correlations between observables that cannot be measured jointly. Here, I explain how the complex valued statistics observed in weak measurements relate to the operator algebra of the conventional Hilbert space formalism and show that the algebra of operators originates from more fundamental relations between the physical properties of a quantum system that can be expressed in terms of complex conditional probabilities. In particular, commutation relations can be identified with fundamental imaginary correlations that characterize the relations between physical properties in terms of their transformation dynamics. Non-commutativity thus originates from a definition of relations between physical properties that replaces the assumption of joint reality with a complex-valued probability reflecting the dynamical response of the system to external forces, e.g. in measurement interactions.